Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly.
翻译:Monte Carlo(PIMC)是利用Markov 链条Monte Carlo(Monte Carlo)从古典Gibbs分布中取样,以估计二次量子旋流系统的热平衡特性的一种方法。PIMC方法被广泛用于研究材料物理学和模拟量排射,但这些成功的应用很少附有正式证明,证明PIMC背后的Markov链条迅速与理想的均衡分布相融合。在这项工作中,我们分析了1D 二次量级汉密尔顿人PIMC的混合时间,包括具有长距离变形相互作用的干扰跨反Ising模型(TIM),以及具有近邻互动的干扰 XY 旋转链。通过将材料的集合时间与均衡分布相连接,我们严格证明使用PIMC 来估计这些模型在反温条件下的分布功能和可观测值的预期值,在大多数对调时,与qubitribits数量相比。混合时间分析基于适用于长距离变形变形变形变形变形模型的单点Monpolpoltical robolov 方法,在Glibislov 的2D Climalismmmissmissmissionalismissional 中,在不为不为常变形变形变型的2-colmocolmismismismolmolmismismismismismismismismmlusismmus 。在不为不为史上,而已知变型号。